Entropic bounds on coding for noisy quantum channels

a r X i v :q u a n t -p h /9707023v 2 31 M a r 1998Entropic bounds on coding for noisy quantum channelsNicolas J.Cerf 1,21W.K.Kellogg Radiation Laboratory,California Institute of Technology,Pasadena,California 911252Information and Computing Technologies Research Section,Jet Propulsion Laboratory,Pasadena,California 91109(Received 9July 1997)In analogy with its classical counterpart,a noisy quantum channel is characterized by a loss ,a quantity that depends on the channel input and the quantum operation performed by the channel.The loss reflects the transmission quality:if the loss is zero,quantum information can be perfectly transmitted at a rate measured by the quantum source entropy.By using block coding based on sequences of n entangled symbols,the average loss (defined as the overall loss of the joint n -symbol channel divided by n ,when n →∞)can be made lower than the loss for a single use of the channel.In this context,we examine several upper bounds on the rate at which quantum information can be transmitted reliably via a noisy channel,that is,with an asymptotically vanishing average loss while the one-symbol loss of the channel is non-zero.These bounds on the channel capacity rely on the entropic Singleton bound on quantum error-correcting codes [Phys.Rev.A 56,1721(1997)].Finally,we analyze the Singleton bounds when the noisy quantum channel is supplemented with a classical auxiliary channel.PACS numbers:03.65.Bz,03.67.Hk,89.70.+cKRL preprint MAP-215I.INTRODUCTIONWithin recent years,the quantum theory of informa-tion and communication has undergone a dramatic evolu-tion (see,e.g.,[1]).Major progress has been made toward the extension to the quantum regime of the classical the-ory of information pioneered by Shannon [2].In particu-lar,the use of quantum communication channels in order to transmit not only classical information but also intact quantum states (or quantum information)has received a considerable amount of attention,following the proof of the quantum analog of Shannon’s fundamental theorem for noiseless coding by Schumacher [3].It has been shown that the von Neumann entropy plays the role of a quan-tum information-theoretic entropy in the sense that it characterizes the minimum amount of quantum resources (e.g.,number of quantum bits)that is necessary to code an ensemble of quantum states with an asymptotically vanishing distortion in the absence of noise.This result suggests that a general quantum theory of information,paralleling Shannon theory,can be developed based on this concept.While such a full theory does not exist as of yet,a great deal of effort has been devoted to this issue over the last few years,and several fundamental results have been obtained,ranging from entanglement-based communication schemes [4]to quantum error-correcting codes [5].In particular,a substantial amount of work has been devoted recently to the transmission of arbitrary states (or quantum information)through noisy quantum channels (see,e.g.,[6–9]).A quantum state processed by such a channel undergoes decoherence by interacting with an external system or environment,which effects an alter-ation of quantum information.A natural question that arises in this context concerns the possibility of transmit-ting quantum information reliably ,in spite of quantum noise,if it is suitably encoded as sequences of quantum bits in analogy with the standard construction used for classical channels.More specifically,a fundamental is-sue is to understand the quantum analog of Shannon’s noisy channel coding theorem and to define the capacity of a noisy quantum channel,i.e.,an upper limit to the amount of quantum information that can be processed with an arbitrarily high fidelity.While several attempts have been made to define a quantum analog of Shannon mutual information that would be a natural candidate for such a quantum measure of capacity (see the concepts of coherent information [7,8]or von Neumann mutual en-tropy [9,10]),the problem of characterizing in general the capacity of a noisy quantum channel is still unsolved.The purpose of this paper is to further clarify the de-scription of noisy quantum channels centered on the von Neumann mutual entropy (see [9]).It has been shown re-cently that a consistent information-theoretic framework that closely parallels Shannon’s construction can be de-veloped,based on von Neumann conditional and mutual entropies [10–13].The central peculiarity of this frame-work is that it involves negative conditional entropies in order to account for quantum non-local correlations between entangled variables.This is in contrast with Shannon information theory in which marginal and con-ditional entropies are all non-negative quantities.Neg-ative quantum conditional entropies simply reflect the non-monotonicity of the von Neumann entropy [14](the entropy of a composite system can be lower than that of its components if the latter are entangled).The resulting information-theoretic formalism provides grounds for the quantum extension of the usual algebraic relations be-tween Shannon entropies in multipartite systems [11–13].Surprisingly,many concepts of Shannon theory can be straightforwardly translated to the quantum regime by extending the range for quantum(conditional and mu-tual)entropies with respect to the classical one in or-der to encompass entanglement[10].This is very help-ful in analyzing quantum information processes in a uni-fied framework,paralleling Shannon theory.For exam-ple,entanglement-based quantum communication pro-cesses[10],quantum channels[9],and quantum error-correcting codes[15]can be described along these lines. In this paper,we focus on the application of this information-theoretic framework to the issue offinding upper bounds on the capacity of quantum codes and quantum channels.In Section II,we outline the general treatment of noisy quantum channels based on quantum entropies[9],and extend it to the characterization of con-secutive uses of a quantum memoryless channel(cf.the notions of one-symbol and average loss explained in Sec-tion IID).This provides a simple framework to consider block coding with quantum channels.Note that,just as in Shannon information theory,quantum entropic consid-erations alone do not result in constructive methods for building codes.Rather,they are useful to derive bounds on what can possibly be achieved or not,from basic prin-ciples.Accordingly,we analyze in Section III several upper bounds(based on the Singleton bound on quan-tum codes[15])for standard quantum channels such as the quantum erasure or depolarizing channel.This con-firms bounds on the quantum capacity that were derived otherwise,but places this problem in a unified context. Finally,we examine in Section IV the extension of this quantum entropic treatment of noisy quantum channels to the case where an auxiliary classical channel is avail-able.Quantum teleportation appears then as a special case of this construction when no block coding is applied.II.ENTROPIC CHARACTERIZATION OFNOISY QUANTUM CHANNELSA.NotationsLet us start by summarizing the basic definitions that will be useful in the rest of this paper when consider-ing noisy quantum channels.The entropy of a quantum system X(of arbitrary dimension)is defined as the von Neumann entropy of the density operatorρX that char-acterizes the state of X,i.e.,S(X)=S[ρX]≡−Tr(ρX log2ρX)(2.1) It can be viewed as the uncertainty about X in the sense that it measures(asymptotically)the minimum number of quantum bits(qubits)necessary to specify X[3].This definition can be extended to the notions of conditional and mutual von Neumann entropies,based on a simple parallel with their classical counterparts which is moti-vated in[10–12].For a bipartite system XY character-ized byρXY,the conditional von Neumann entropy isS(X|Y)=S(XY)−S(Y)(2.2) while the mutual von Neumann entropy isS(X:Y)=S(X)−S(X|Y)=S(Y)−S(Y|X)=S(X)+S(Y)−S(XY)(2.3) where S(XY)is calculated fromρXY,while S(X)and S(Y)are obtained from the reduced density operatorsρX=Tr Y(ρXY)andρY=Tr X(ρXY).Subadditiv-ity of quantum entropies implies S(X:Y)≥0,where the equality holds if X and Y are independent(i.e.,ρXY=ρX⊗ρY).Note that,when S(XY)=0(i.e.,the joint system XY is in a pure state),we have S(X:Y)=2S(X)=2S(Y)as a consequence of the Schmidt decom-position.This property will be useful in the following. Several quantum entropies can also be defined for char-acterizing multipartite quantum systems.Consider,for instance,a tripartite system XY Z.The von Neumann conditional mutual entropy(of X and Y,conditionally on Z)can be defined asS(X:Y|Z)=S(X|Z)−S(X|Y Z)=S(X|Z)+S(Y|Z)−S(XY|Z)=S(XZ)+S(Y Z)−S(Z)−S(XY Z)(2.4) in perfect analogy with the classical expressions.Note that the strong subadditivity of quantum entropies im-plies S(X:Y|Z)≥0[12].We can also define the von Neumann ternary mutual entropy asS(X:Y:Z)=S(X:Y)−S(X:Y|Z)(2.5) Note that,if S(XY Z)=0(i.e.,the ternary system is in a pure state),then S(X:Y:Z)=0[12],or,equivalently, S(X:Y)=S(X:Y|Z),a property which is very useful in the analysis of quantum channels.Also,chain rules for quantum entropies can be written,such asS(X:Y Z)=S(X:Y)+S(X:Z|Y)(2.6) which parallel the classical relations[12].The motiva-tion for building such a quantum entropic framework is that it provides an information-theoretic formulation of quantum entanglement in multipartite systems,unified with Shannon’s description of classical correlation.It is an extension of Shannon’s formalism beyond its original range,as reflected for example by the fact that the quan-tum mutual entropy can reach twice the maximum value allowed for classical entropies[10],that is,0≤S(X:Y)≤2min[S(X),S(Y)](2.7) This factor2appears in many quantum information-theoretic relations(see below),and originates from the Araki-Lieb inequality for quantum entropies[10–12].B.Quantum mutual entropy,loss,and noiseLet us now outline the entropic treatment of a noisy quantum channel(see also Ref.[9]).Such a treatmentexplicitly displays the correspondence with the standard description of noisy classical channels(see Appendix A),thereby unifying classical and quantum channels.Ourdescription involves three quantum systems of arbitrary dimensions:Q(the quantum system whose processingby the channel is concerned),R(a“reference”systemwhich Q is initially entangled with),and E(an exter-nal system or environment which Q is interacting within the noisy channel).More specifically,we assume that Q is initially entangled with R,so that the joint stateof Q and R is the pure state|ΨRQ .We may as well regard Q as a quantum source,being initially in a mixed stateρQ(realized by a given ensemble of quantum statesassociated with some probability distribution).The“pu-rification”ofρQ into|ΨRQ can always achieved by ex-tending the Hilbert space H Q to H RQ,so that we have ρQ=Tr R(|ΨRQ ΨRQ|).The corresponding reduced von Neumann entropies areS(R)=S(Q)≡S(2.8) where S is called the source entropy.In the dual picture where an arbitrary pure state of Q(rather than entan-glement)is sent through the channel,S then measures the“arbitrariness”of Q(it can be viewed as the aver-age number of quantum bits that are to be processed by the channel in order to transmit the state of Q).In what follows,we prefer to consider a quantum input Q that is entangled with R,so that the preservation of entanglement—rather than of arbitrary states—will be the central feature of a quantum transmission channel. The initial mutual entropy to be transmitted is thusS(R:Q)=2S(2.9) that is,twice1the source entropy.When it is processed by the channel,Q interacts withE(assumed to be initially in a pure state|0 )according to the unitary transformation U QE,inducing decoherence. This describes the most general(trace-preserving)oper-ation of a quantum channel that is allowed by quantum mechanics.Roughly speaking,the resulting noisy quan-tum channel is such that,typically,only a fraction of the initial entanglement with R can be recovered after having been processed by the channel(the rest of the entangle-ment with R is lost,in the sense that it is transferred to the environment).More specifically,the decohered quan-tum system after interaction with E,denoted as Q′,is in the stateρ′Q=Tr E U QE(ρQ⊗|0 0|)U†QE (2.10)whereρQ is the initial state of Q(with source entropy S).The completely positive linear mapρQ→ρ′Q cor-responds to the“quantum operation”performed by the noisy channel[7].After such an environment-induced decoherence,the joint system R′Q′E′is in the state |ΨR′Q′E′ =(1R⊗U QE)|ΨRQ |0E whose entropy Venn diagram is represented in Fig.1(the primes refer to the systems after decoherence).Note that,as the reference is not involved in decoherence,we have R′≡R.FIG.1.Schematic representation of the quantum opera-tion effected by a noisy quantum channel.The quantum sys-tem Q is initially entangled with the reference R,with a mu-tual entropy of twice the source entropy S(this is indicated by a dashed line).Then Q decoheres by interacting with an environment E(initially in a pure state|0 ).The entropy Venn diagram summarizes the entropic relations between Q′(output of the quantum channel),R′(reference),and E′(en-vironment)after decoherence.The three parameters,I,L, and N,denote the von Neumann mutual entropy(quantum information),the loss,and the noise,respectively.QR2SR’Q’E’-I/2-L/2-I/2-N/2-N/2-L/2N LR’Q’E’IThe entropy diagram of R′Q′E′depends on three pa-rameters,the von Neumann mutual entropy(or the quan-tum information)I,the loss L,and the noise N,these quantities being defined in analogy with their classical counterparts:I=S(R:Q′)(2.11)L=S(R:E′|Q′)=S(R:E′)(2.12)N=S(Q′:E′|R)=S(Q′:E′)(2.13)The classical correspondence can be made fully explicit by including an environment in the description of a clas-sical channel,as shown in[9].The second equality inEqs.(2.12)and(2.13)has no classical analog,and re-sults from the vanishing of the ternary mutual entropy S(R:Q′:E′)(see[9,15]).Physically,the quantum infor-mation I corresponds to the residual mutual entropy be-tween the decohered quantum output Q′and the refer-ence system R that purifies the quantum input Q.The loss L is the mutual entropy that has arisen between the environment after decoherence E′and the reference sys-tem R,while the noise N is the mutual entropy between the decohered quantum output Q′and the environment E′.Note that I,L,and N can be written as a function of reduced entropies only,without explicitly involving the environment E in the discussion,by making use of the Schmidt decomposition of the state of R′Q′E′,namely S(E′)=S(RQ′):I=S(Q)+S(Q′)−S(RQ′)(2.14)L=S(Q)+S(RQ′)−S(Q′)(2.15)N=S(Q′)+S(RQ′)−S(Q)(2.16) It can also be shown that these three quantities are in fact independent of the choice of the reference system R whenever the latter purifies the quantum input Q,so that they provide a most concise entropic characterization of informationflow in the channel.They depend in gen-eral on the channel input(i.e.,ρQ)and on the quantum operation performed by the channel(i.e.,the completely positive trace-preserving map on Q that is specified by U QE in the joint space of Q and E).This exactly paral-lels the situation for the analog classical quantities.The information I,loss L,and noise N of a classical channel of input X and output Y(see Appendix A)indeed de-pend on the input distribution p(x)and on the channel “operation”characterized by p(y|x).Among these three quantities,only I and L are rele-vant as far as(forward)information transmission through the channel is concerned(the noise N plays a role in the description of the“reverse”channel,just as for classical channels).Indeed,information processing is character-ized by the balance between the von Neumann mutual entropy and the loss,these two quantities always sum-ming to twice the source entropy:I+L=2S(Q)≡2S(2.17) The mutual entropy I=S(R:Q′)represents the amount of the initial mutual entropy with respect to R(i.e.,2S) that has been processed by the channel,while the loss L=S(R:E′)corresponds to the fraction of it that is unavoidably lost in the environment.If the channel is lossless(L=0),then I=2S,so that the interaction with the environment can be perfectly“undone”,and the initial entanglement of Q can be fully recovered by an appropriate decoding[7,9].(Equivalently,this means that an arbitrary initial state of Q can be recovered with-out error.)This can be understood by noting that R does not become entangled directly with the environment in a lossless channel,but only via the output Q′(see Fig.1 when L=0).An operation on Q′only(namely,the de-coding operation)is enough to transfer the entanglement with E′(measured by the noise N)to an ancilla,while preserving the entanglement2S with R.Thus,if L=0,a perfect transmission of informa-tion(including quantum information)can be achieved through the channel by applying an appropriate decod-ing.When I=0,on the other hand,no information at all(classical or quantum)can be processed by the channel.This is the case,for example,of the quantum depolarizing channel with p=3/4(see Section IIID).In between these limiting cases,classical information(and, up to some restricted extent,quantum information)can be reliably transmitted at the expense of a decrease in the rate by making use of block coding.The analysis of such a transmission of quantum information immune to noise is the main focus of this paper.For completeness,let us mention that a channel with N=0is the quantum analog of a deterministic chan-nel[16],that is,a channel where the input fully deter-mines the output(see Appendix A).The quantum output Q′is indeed not directly entangled with E′but only via R,which implies that its entanglement with R remains intact(see Fig.1when N=0).This does not mean, however,that perfect error correction is achievable,as an operation on the reference R is needed to recover the initial entanglement2S between Q and R.A channel which is both lossless(L=0)and deterministic(N=0) is called noiseless;its action on Q is the identity operator (or anyfixed unitary operator).For example,the overall channel including a noisy quantum channel along with the encoder and decoder is obviously noiseless if perfect error correction is achieved.(In other words,the decoder is used to eliminate the quantum noise N=0by trans-ferring the entanglement with E to an ancilla,which then makes the overall channel noiseless provided that L=0.) It is worth noting here that the noise N and the loss L play symmetric roles when considering the“reverse”channel obtained by interchanging the input and output. (This is true for classical channels as well.)More specif-ically,N and I always sum to twice the output entropy,I+N=2S(Q′)(2.18) in analogy with Eq.(2.17).Roughly speaking,N plays the role of the loss of the reverse channel,as shown in Sec.IIC.C.Properties of quantum I,L,and NThe above entropies for a noisy quantum channel can be shown to fulfill several properties,akin to classical ones,which make them reasonable quantum measures of information,loss,or noise(see also Ref.[9]).First,thequantum mutual entropy I can be shown to be concave in the inputρQ for afixed channel,i.e.,afixed quantumoperationρQ→ρQ′or afixed U QE.Therefore,any lo-cal maximum of I is the absolute maximum,that is,thevon Neumann capacity of the channel.This parallels the concavity of the Shannon mutual entropy H(X:Y)in the input probability distribution p(x)for afixed channel, i.e.,fixed p(y|x)[17].Second,I is convex in the output ρQ′for afixed inputρQ.This property will be used in the next Section when considering a“probabilistic”channel (the effective channel resulting from the probabilistic use of a family of channels).It is the quantum analog of the property that the information H(X:Y)processed by a classical channel is a convex function of p(y|x)for afixed p(x)[17].These two properties are simple to prove by reexpressing the von Neumann mutual entropy I asS(R:Q′)=S(Q′E′)+S(Q′)−S(E′)=S(Q′)+S(Q′|E′)(2.19) or asS(R:Q′)=S(R)+S(Q′)−S(RQ′)=S(R)−S(R|Q′)(2.20) If the inputρQ is a convex combination of density opera-tors while the channel isfixed,it is easy to see thatρQE and thereforeρQ′E′are also convex combinations(as the channel operation is linear).Since the conditional en-tropy S(Q′|E′)is concave in a convex combination of ρQ′E′while S(Q′)is concave inρQ′[14],Eq.(2.19)im-plies the concavity of the quantum mutual entropy I in the input for afixed channel.The second property can be proven the same way by noting that,if we have a “probabilistic”channel—a convex combination of quan-tum channels—acting on afixed input,thenρRQ′is a convex combination of density operators whileρR is con-stant.Thus,Eq.(2.20)together with the concavity of the conditional entropy S(R|Q′)in a convex combina-tion ofρRQ′implies that the quantum mutual entropy I is convex in the output for afixed input.A third important property is that the mutual entropyI and the quantum loss L are subadditive when consid-ering a channel made of several independent quantum channels used in parallel.This will be shown when ana-lyzing quantum block coding(cf.Sect.IID).Finally,it can be proved that I obeys(forward and reverse)data-processing inequalities when considering chained quan-tum channels.If we chain two channels by using the output of thefirst as an input for the second(see Fig.2), the total(1+2)channelρQ→ρQ′→ρQ′′is characterized byI12=S(R:Q′′)(2.21)L12=S(R:E′E′′)(2.22)N12=S(Q′′:E′E′′)(2.23)since we can regard the two environments E′and E′′as a global environment for this total channel.FIG.2.Schematic view of the chaining of two noisy quan-tum channels.In each of them,the input state decoheres by interacting with a(separate)environment.The input of the first channel is initially entangled with R,with a source en-tropy of S(see the dashed line).The output of this channel Q′is then used as an input for the second channel.Since Q′is purified by RE′(not by R alone),the“reference”system that must be considered in the entropic characterization of the second channel is RE′.’’’R0E02SQ12Q’E’’QUsing the chain rule for quantum mutual entropies S(R:E′E′′)=S(R:E′)+S(R:E′′|E′),and remembering that S(R:E′′|E′)≥0as a result of strong subadditivity, we obtain0≤L1≤L12(2.24) where L1=S(R:E′)is the loss of thefirst channel while L12is the loss of the total channel.Thus,the loss can only increase by further processing of quantum informa-tion in the second channel.Since I1+L1=2S(Q)and I12+L12=2S(Q),we obtain the forward data-processing inequalityI12≤I1≤2S(Q)(2.25) implying that the mutual entropy of the total channel cannot exceed the one of thefirst channel.This is the quantum analog of H(X:Z)≤H(X:Y)≤H(X)for chained classical channels X→Y→Z[17].Now,if we use the chain rule S(Q′′:E′E′′)= S(Q′′:E′′)+S(Q′′:E′|E′′)together with strong subad-ditivity,we obtain0≤N2≤N12(2.26) where N2=S(Q′′:E′′)is the noise of the second chan-nel while N12is the noise of the total channel.As I2+N2=2S(Q′′)and I12+N12=2S(Q′′),we obtain the reverse data-processing inequalityI12≤I2≤2S(Q′′)(2.27) where I2=S(RE′:Q′′)is the mutual entropy processed by the second channel.(Note that the“reference”sys-tem that purifies the input Q′of the second channel is RE′.)This parallels the classical inequality H(X:Z)≤H(Y:Z)≤H(Z)for chained channels[17].Eqs.(2.24) and(2.26)emphasize that the loss L and the noise Nplay a symmetric role in this entropic description if one interchanges the input and the output of the quantum channel(“time-reversal”),just as for classical channels. This is reflected by the symmetry between the forward and the reverse data-processing inequalities.D.One-symbol loss and average lossThe central idea of classical error correction by block coding is to introduce correlations between the bits that make a block,in order to have redundancy in the trans-mittedflow of data.This can make the transmission asymptotically immune to errors,up to some level of noise.In quantum error-correcting codes,the qubits that form a block are entangled in a specific way,so that a par-tial alteration due to decoherence can be recovered[5]. Even though entanglement gives rise to some qualita-tively new features(see[15]for a detailed analysis),the objective is ly,when block coding is used, i.e.,when say k“logical”qubits are encoded into blocks of n“physical”qubits,it is possible to achieve a situation where the overall loss of the joint(n-bit)channel is arbi-trarily small,while the loss for individual qubits(for each use of the channel)isfinite.In analogy with the classi-cal construction,if blocks of n qubits that are initially entangled with respect to R(with a mutual entropy2k) can be transmitted through the channel with an asymp-totically vanishing overall loss,we say that the channel processes2k/n bits of entanglement per qubit.Equiva-lently,the channel is transmitting at a rate R=k/n(on average,k arbitrary binary quantum states can be trans-mitted for n transmitted qubits).The maximum rate at which quantum information can be reliably sent through the noisy channel is defined as the quantum channel ca-pacity.(This maximum has to be taken over all possible coding schemes,and for n→∞.)Whether a good(and operational)definition of such a“purely quantum”chan-nel capacity exists is currently an open question.In the following,we restrict ourselves to the issue offinding up-per bounds on the rate of perfect quantum information transmission(and therefore on such a“purely quantum”capacity).Let us consider the asymptotic use of a quantum dis-crete memoryless channel,where n(tending to infin-ity)qubits are transmitted sequentially.2Each qubit may decohere due to an environment(quantum noise), the exact interaction depending on the considered noise model.The important point is that the environment foreach qubit is initially independent of the one interact-ing with every other qubit.Thus the information pro-cess can be viewed as n sequential uses of a quantum memoryless channel(the environment being“reset”af-ter each use)or,equivalently,as n parallel independentchannels processing one qubit each(see Fig.3).We as-sume that the set of n input symbols(Q1,...Q n)areinitially entangled with R,so that S(R:Q1···Q n)=2S and S(R)=S(Q1···Q n)=S.If we consider these n symbols as the single input of a joint n-bit channelQ1···Q n→Q′1···Q′n,information transmission is de-scribed by the mutual entropyI=S(R:Q′1···Q′n)=S(Q1···Q n)+S(Q′1···Q′n)−S(E′1···E′n)=S(Q′1···Q′n|E′1···E′n)+S(Q′1···Q′n)(2.28) and the lossL=S(R:E′1···E′n)=S(Q1···Q n)+S(E′1···E′n)−S(Q′1···Q′n)=S(E′1···E′n|Q′1···Q′n)+S(E′1···E′n)(2.29) where we have made use of the conservation of entropy imposed by the unitarity of the global interaction with the n environments E1,...E n.Obviously,we have I+L=2S(Q1···Q n)=2S(R),which is twice the source entropy S of the joint channel.FIG.3.Schematic view of a memoryless quantum channel.This channel is used n times,but the environment is“reset”after each use.This can be viewed as n parallel(independent) channels,each one being used for one of the input symbols. The n input symbols(Q1,Q2,···Q n)are initially entangled with R(as indicated by a dashed line),with a joint source entropy of S.1’Q2SRQQn2’’2n1QQQ1’’’EEE2n。